It's not really that the space of locally constant, compactly-supported (complex-valued) functions on a p-adic (or other totally disconnected) group has no topology. Rather, it has a canonical topology with some convenient features. That is, that space is a "strict" (filtered) colimit of finite-dimensional spaces. That is, it is a countable ascending union of finite-dimensional spaces. Finite-dimensional spaces have unique topological vector space topologies. This colimit has a unique (topological vector space) topology. The reason that for many purposes we can ignore the topology is that every linear map from that space to any other topological vector space is continuous: this follows from the corresponding fact for finite-dimensional TVS's and the definition of "colimit".

As in the question, there are times when the topology matters to some degree. Happily, it is quasi-complete (despite not being complete-metric), which is sufficient for various version of vector-valued integrals to make sense, whether Bochner-style integrals or Gelfand-Pettis-style.

As an easier case, it is standard that a compactly-supported, continuous, $V$-valued function (for $V$ locally convex, quasi-complete) $F$ has an integral with expected properties. As with L. Schwartz' treatment of "Schwartz functions" as extending to a suitable one-point compactification, if/when we can compactify the physical set on which we integrate, then we can immediately apply this simplest case of vector-valued integration.

(Of course, the integrals defining these intertwinings only converge for parameters in some cone, and must be meromorphically continued...)

It's not really that the space of locally constant, compactly-supported (complex-valued) functions on a p-adic (or other totally disconnected) group has no topology. Rather, it has a canonical topology with some convenient features. That is, that space is a "strict" (filtered) colimit of finite-dimensional spaces. That is, it is a countable ascending union of finite-dimensional spaces. Finite-dimensional spaces have unique topological vector space topologies. This colimit has a unique (topological vector space) topology. The reason that for many purposes we can ignore the topology is that every linear map from that space to any other topological vector space is continuous: this follows from the corresponding fact for finite-dimensional TVS's and the definition of "colimit".

As in the question, there are times when the topology matters to some degree. Happily, it is quasi-complete (despite not being complete-metric), which is sufficient for various version of vector-valued integrals to make sense, whether Bochner-style integrals or Gelfand-Pettis-style.

As an easier case, it is standard that a compactly-supported, continuous, $V$-valued function (for $V$ locally convex, quasi-complete) $F$ has an integral with expected properties. As with L. Schwartz' treatment of "Schwartz functions" as extending to a suitable one-point compactification, if/when we can compactify the physical set on which we integrate, then we can immediately apply this simplest case of vector-valued integration.

(Of course, the integrals defining these intertwinings only converge for parameters in some cone, and must be meromorphically continued...)

EDIT: to be clear(er), as in comments, no, indeed, the functions in the induced repn are not compactly supported (although compactly supported mod $P$), but they are completely determined by their values on the maximal compact, and this gives the finite-dimensional feature... But/and one easily finds in the literature misleading remarks about spaces of test functions on t.d. groups "having no topology" or "having discrete topology", and I intended partly to address that.